Hardy spaces Hp over non-homogeneous metric measure spaces and their applications
Let (X,d,μ) be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions. Let ρ∈(1,∞), 0<p≤1≤q≤∞, p=q, γ∈[1,∞) and ε∈(0,∞). In this article, the authors introduce the atomic Hardy space Hatb,ρp,q,γ(μ) and the molecular Hardy space Hmb,ρp,q,γ,ε(μ) via the discrete coefficient KB,S(ρ),p, and prove that the Calderón-Zygmund operator is bounded from Hmb,ρp,q,γ,δ(μ) (or Hatb,ρp,q,γ(μ)) into Lp(μ), and from Hatb,ρ(ρ+1)p,q,γ+1(μ) into Hmb,ρp,q,γ,21(δ−pν+ν)(μ) whose genealized fractional versions are also obtained. The authors also introduce the ρ-weakly doubling condition, with ρ∈(1,∞), of the measure μ and construct a non-doubling measure μ satisfying this condition. If μ is ρ-weakly doubling, the authors further introduce the Campanato space Eρ,η,γα,q(μ) and show that Eρ,η,γα,q(μ) is independent of the choices of ρ, η, γ and q; the authors then introduce the atomic Hardy space Hatb,ρp,q,γ(μ) and the molecular Hardy space Hmb,ρp,q,γ,ε(μ), which coincide with each other; the authors finally prove that Hatb,ρp,q,γ(μ) is the predual of Eρ,ρ,11/p−1,1(μ). Moreover, relations of these Hardy spaces are also discussed.